The Periodic Boundary Value Problem for the Weakly Dissipative -Hunter-Saxton Equation

Abstract

We study the periodic boundary value problem for the weakly dissipative -Hunter-Saxton equation. We establish the local well-posedness in Besov space , which extends the previous regularity range to the critical case.

1. Introduction

In this paper we consider the weakly dissipative -Hunter-Saxton (HS) equation with periodic boundary condition [1]:where , , , and the term models energy dissipation.

When and , (1) is the Hunter-Saxton equation [2]The Hunter-Saxton (HS) equation is an asymptotic equation for rotators in liquid crystals and modeling the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal [2]. The HS equation has a bi-Hamiltonian structure [3, 4] and is completely integrable [5]. The initial value problem of the HS equation also has been studied extensively; see [2, 6, 7]. We refer to [8–12] for the weak solution to the HS equation. When , the global existence and blow-up phenomena for the dissipative HS equation can be found in [13].

When , (1) becomes the -HS equationwhich is derived and studied in [14]. Khesin et al. [14] showed that the -HS equation describes the geodesic flow on with the right-invariant metric given at the identity by the inner productMoreover, if interactions of rotators and an external magnetic field are allowed, then the -HS equation can be viewed as a natural generalization of the rotator equation. The periodic peaked solution and the multipeakons solution to -HS equation were showed in [14, 15].

Another important model which possesses some similar structures to the -HS equation is the Camassa-Holm (CH) equation:The Camassa-Holm (CH) equation was derived independently by Fokas and Fuchssteiner in [16] and by Camassa and Holm in [17]. Fokas and Fuchssteiner derived (6) in studying completely integrable generalizations of the KdV equation with bi-Hamiltonian structures, while Camassa and Holm proposed (6) to describe the unidirectional propagation of shallow water waves over a flat bottom. In [18], Constantin and Lannes proved that the CH equation, as a model for the propagation of shallow water waves, is valid approximations to the governing equations for water waves. The relation between the CH and HS equation is that the HS equation can be viewed as the high-frequency limit of the CH equation [5]. In [17], Camassa and Holm discovered the bi-Hamiltonian structure of CH, which ensures the existence of an infinite number of conservation laws. The integrability of CH (as an infinite-dimensional Hamiltonian system) was studied in [19, 20]. Moreover, the CH equation is such an equation that exhibits both phenomena of soliton interaction (peaked soliton solutions) and wave breaking (the solution remains bounded while its slope becomes unbounded in finite time [21]). The global existence of strong and weak solution and blow-up phenomena for the CH equation can be found in [21–25].

In general, it is almost impossible to avoid energy dissipation in the real world. Therefore, it is reasonable to study the model with energy dissipation. For example, the weakly dissipative CH equation and weakly dissipative DP equation have been studied in [26] and [27–29], respectively.

Recently, Liu and Yin [1] suggested the weakly dissipative -HS equation. They established the local well-posedness in Sobolev spaces , obtained the global existence, and studied the blow-up scenario. Motivated by the previous work, in this paper, we aim to prove the local well-posedness for (1) (2) in the critical Besov space .

Integrating (1) with respect to yields , which implies thatNow we denote and reformulate (1), (2) as the nonlocal form:where withFor the inverse operator , the relation is given explicitly by [14, 30], In terms of Green’s function, we see that can be also expressed as [14, 30] where is the periodic extension of with , Besides, as and commute, then We will show that the nonlocal terms map to (see Lemma 10) and therefore we can expect the local well-posedness of (1) in .

Notations. We use , , and to denote estimates that hold up to some universal constant. is also a generic constant that may assume different values in different lines. is the space of all infinitely differentiable functions on and is its dual space. All function spaces are over and we drop in our notations of function spaces if there is no ambiguity.

The main results of this paper are as follows.

Theorem 1. When , we have the following results:(1)There exists a such that (1), (2) have a unique solution which depends continuously on the initial data . Furthermore, for some , we have(2)If , are the solutions with initial data , respectively, then where with the constant depending on the radius and . That is to say, the solution map defined by (1), (2) is Hölder continuous as a map from , with the topology, to .

Now we outline the proof of Theorem 1. We construct a sequence of the approximate solutions . We need to prove that is a Cauchy sequence. However, in the critical case, the following estimate cannot be obtained due to the lack of appropriate product estimate in (this is one of the significant differences between the critical case and the supercritical case ). Therefore we will use a product estimate in obtained in Danchin’s cerebrated paper [31] (see Lemma 7) and the Log interpolation inequality (see Lemma 8) to find Then we prove the convergence of in firstly and then to obtain the convergence in by interpolation. To this end, we will use the Osgood lemma (Lemma 9) to estimate . Our idea is motivated by [32, 33], where the authors used the Osgood lemma to studied the b-family equation and the modified Camassa-Holm equation, respectively.

The remainder of this paper is organized as follows. In the next section, we state some preliminaries. We prove Theorem 1 in Section 3.

2. Preliminaries

Now we state some preliminaries on the Littlewood-Paley theory. We refer to [34–36] for the elementary properties of them.

Let , be two smooth radial functions satisfying , , , , and

We decompose into Fourier series; that is, , where is the Fourier transform and the inverse transform is given by . Now we define the periodic dyadic blocks as

Then we define the low frequency cut-off as Direct computation implies that, for any , we have the quasi-orthogonality properties: Furthermore, for all , and

Definition 2 (Besov spaces). Let , , and . The nonhomogeneous Besov space is defined by where .

The following lemma summarizes some useful properties of .

Lemma 3 (see [34–36]). Let and , , . Then the following properties hold:(1)Consider if ; consider and .  is locally compact if ; .(2), is a Banach algebra. is a Banach algebra (or and ).(3)Consider , , and(4)Consider , , , and (5)If is bounded in and converges to in , then and

For , we see that , where if and if . Therefore, we see that if , with , then , which means that is bounded from into .

Now we recall some results of the transport equation (see [34] for the details).

Lemma 4 (a priori estimates). Let , , and Assume that , , and if or otherwise. If solves the 1D linear transport equationthen there exists a constant depending only on , , such that the following statements hold:(1) For all , and hence where (2) If , then holds true for all with

Lemma 5 (existence and uniqueness). Let , , , , and be as in the statement of Lemma 4. Assume that for some , , and if or , , and if . Then (24) have a unique solution for any and the inequalities of Lemma 4 hold true. Moreover, if , then .

We need the following lemmas.

Lemma 6 (Moser-type estimates; see [34, 37]). Let , , and ; then

Lemma 7 (see [34]). For any , , there holds the product estimate

Lemma 8 (see [34]). There is a constant such that, for , , and ,

Lemma 9 (Osgood lemma; see [34, 38]). Let be a measurable function, a locally integrable function, and a continuous and increasing function. Assume that, for some nonnegative real number , the function satisfies If , then with .
If and verifies the condition , then the function .

Now we establish some estimates on and the difference of two solutions.

Lemma 10. Let be given in (9); then there holds the estimate:

Proof. Since , (32) follows from Lemma 6 easily and we omit the proof here.

Lemma 11. For any , in , the following estimates hold:

Proof. Equation (33) follows from (9) and Lemma 7. We omit the details here.

Lemma 12. If are two solutions to (1) for some with initial data , respectively, then, for any and ,where is a constant depending on and .

Proof. Since , are two solutions to (1) with initial data , respectively, we know that satisfieswhere is given in (9). Let , by Lemma 4; we have where . Since , we see thatand therefore we have Since , by Lemma 11 and (37), we arrive at Hence . From Lemma 8 (, ) and (37), we obtain that Consequently, for all ,Using the fact that , we can infer from (40) that Thanks to Lemma 9, we obtain Therefore, we haveIf , by interpolation, the embedding , and (37), we obtain where . Using (43), we obtain thatWe finish the proof by combining (43) and (45).

Lemma 13 (see [31]). Denote Let be a sequence of continuous bounded functions on with for some and . Assume that solves with , . If in , then in

3. Proof of Theorem 1

In this section, we prove Theorem 1 by the following several steps.

3.1. Approximate Solutions and Their Uniform Bounds

Starting from and by induction, we define a sequence of smooth functions by solving the following transport equation iteratively:where and given in (9). Since all the data belong to , from Lemma 5 and by induction, we can show that, for all , the above equation has a global solution belonging to For , set . Since , from the estimate (26) of Lemmas 4 and 10, we haveReset the constant on the above estimate such thatLet such that ; we claim thatAssume (50) is true for . We now prove that it also holds true for . Actually, by (50), we haveFrom the above equation, we see clearly that when , , we haveCombining (49), (50), (51), and (52) gives rise to Hence (50) is true for . Setting , for , we can conclude that exists for and satisfies the boundFurthermore, for , using (54) yieldsHence we conclude that is uniformly bounded.

3.2. Convergence of the Approximate Solutions

We now prove that is a Cauchy sequence in . Firstly, we see that the estimate of is essential in the derivation of Lemma 12. Similar to Lemma 12, we obtain that, for ,where we used the following estimate and we have (see, e.g., in [39], page 2142) Let . Since is uniformly bounded, from (56), we obtain that As the function is nondecreasing, we see that satisfies Let . For any given , there exists an such that when , we have , from which it follows that Hence we have Since is arbitrary, we have Since is nondecreasing and , we can infer from Lemma 9 that ; in other words, is a Cauchy sequence in . By (5) in Lemma 3, we see Hence is actually Cauchy sequence in and therefore converges to some function .

3.3. Existence, Regularity, and Uniqueness of the Solution

Since is uniformly bounded by , property (5) in Lemma 3 guarantees that , which means that . If , we haveIf , by interpolation again, we havewhere . From (64) and (65), we see that in for all . Taking limits to (47), we can deduce that indeed solves (8). Since , from Lemmas 5 and 10, we know that . By (8), we obtain that . The uniqueness is a corollary of Lemma 12.

3.4. Hölder Continuity of the Solution Map

The continuity of the solution map can be obtained by following the steps in [31] and using Lemma 13 and we omit the details in this paper. Now we consider the Hölder continuity of the solution map. For the initial data , by (13), we see that the lifespan of the corresponding solution to (8) satisfies where does not depend on . Therefore, we can find such that, for all , the corresponding solution . Directly from (13) and Lemma 12, (14) is proved.

We complete the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundations of China (no. 11401096) and Guangdong Province (no. 2013KJCX0189 and no. 2014KZDXM063). The authors thank the editors for their hard working and also gratefully acknowledge helpful comments and suggestions by reviewers.